The Art of Proof

Seems like an excellent companion to either M208 or M381 to me. Check out the site of Springer if you are interested in books like this.

Infinite series ( and Euler's identity revisited )

The importance of infinite series and sequences can not be underestimated in my opinion, they literally pop up everywhere. To my surprise however, there isn't a single course at the Open University ( or any other university to my knowledge ) that deals exclusively with the topic of 'Infinite Series'. Instead the theory is stuffed away somewhere else, as if it is not really important. Like in M208 for example. - I have been searching for books on the subject and there -is- a ( recent ) book on the subject. It is called 'Real Infinite Series' by D. Bonar / M. Khoury published my TMAA. Tests not in M208 but in this book are for example Raabe's Test, Rummer's Test. Cauchy's Condensation Test, Abel's Test, and Dirichlet's Test as well as Bertrand's Test. It includes an entire chapter on the harmonic series with different divergent proofs. In the appendix there is an overview of the literature on infinite series.

P.S.
This video shows the proof of $e^{i\pi}+1=0$ using infinite series.

M208 result is in

Grade 2 pass. - More later...

M203/M208 Exam Results 2006-2009

M203/M208

Year	Took exam	Pass
2006	69%	92%
2007	71%	90%
2008	69%	93%
2009	67%	92%

Took exam: Exam candidates / Final Registrations
Pass: Candidates passed / Exam candidates

Source: Open University

Touching complex analysis

Prof. Mattuck is in top form in lecture 6 when he talks about Euler and the beauty of complex numbers. Although it is a lecture in the DE series it can be watched as a stand-alone lecture. So if you are doing MST121, MS221 or M208 it is great fun to watch this lecture. He even touches the field of Complex Analysis when he explains differentiating $e^{i\theta}$. He notes ( jokes ) that time is always a real variable but he isn't so sure when the next Einstein comes around: he may very well decide we need complex time! Anyway, how would you integrate $\int{e^{x}}\cos{x}\ dx$ ? Prof. Mattuck says these integrals are easy if you switch to the complex domain. Finally he solves the beautiful equation $x^n+1=0$ ( as we have seen in MS221, M208 ).



Hopefully, after or during MST209, I will be able to analyze the synchronization of metronomes problem using differential equations some day.

M208 is done.

A rather dark side of my personality manifested itself the last two months or so. Don't worry I killed the monster. I sort of digressed to the period when I was in high school, i.e. I did not send my work on TMAs 6 and 7 to my tutor since I more or less had it with that reptilian. Goodbye wished for, hoped for distinction. More about this in another post. In this post I'll report about the exam itself.

The exam took place in The Hague. I counted four people coming in doing M208 but there could have been more. In the examination room with about 16 or so people I have seen at least three different sets of papers of which S320 was one. It was mentioned that two people did not show up. When we entered the room we got our papers and could choose a place to work for ourselves.

At 2.30 PM ( local time ) we were allowed to open the paper. Same format as MS221, i.e. two parts A and B. Part A had 12 questions with a maximum score of 70. And Part B was a 2 out of 5 set. We could choose two questions of 15 points each. From what I recall the questions in part A were about ( not in order ):
1. Graph of (2x+3) / (3-x), incl. asymptotes and axes-intercepts.
2. Solving an inequality.
3. Diagonalizing a matrix using eigenvectors.
4. Question about [{1,2,4,9,10,12}, mult mod 13]
5. Question about Homomorphism z->|z|^2.
6. Question involving R2 geometry using vectors and inner product
7. Question on series
8. Finding an integral
9. Question on a Taylor series
10. Question about permutations
11. Question about the symmetry group of the pentagon.
12. ( Forgotten )

At 4.30 I had completed 9 out of 12 questions. Last year with MS221 I continued working on part A with the result that I only completed half of a question of part B. I think it was a correct decision. I had three questions left which would take me at least 40 minutes, leaving 20 minutes for part B. I could score 18 max. Continuing with part B at this point gave me an opportunity to collect 30 points although part B questions are somewhat more difficult.
I was getting tired and somewhat stresssed. Again, I did not complete all 12 questions in 2 hours. I did not take time to read all five questions. I started to work on question 13 which was on group theory.

13. A question about some finite group with 16 elements. The Cayley Table was given, nothing else.
- Find a cyclic subgroup, call it H.
- Prove that H is normal and that group K ( given ) is not normal.
- List elements of quotient qroup G/H
- Determine the structure of the group.
- ( One more question, forgotten )

It was now 17.00, I chose the next question on Linear Algebra
14. About a linear transformation in R3.
- Find the dimension of the kernel
- What is the geometry of the kernel
- Find a basis of the image
- What is the geoemtry of the image and find an equation
- Given two sets of 3equations with 3 unknowns determine the number of solutions

I feel about the same as last year with MS221. I am fairly sure ( 99% ) of a pass.

All in all, I have learned a lot this year. About mathematics of course, but also about myself and about the effect a study like this can have on a person. Not everything has been said about this study year. More next time.

M208 Exam

I am off to the exam.

Revision time.

M208 exam is 13-oct-2010 14:30 - 17:30.

Result M208 - TMA05

I scored 75. 11/20 on question 6. I used a methoud using indirect symmetries. Works just as well. I had 18 bricks as an answer -of course-. It is an abstract combinatorial counting problem.

I very much doubt if the person who is tutoring me on M208 really 'owns' the materials or is merely pretending. I suspect the last so a discussion won't work. I haven't got a leg to stand on if I don't score high in the nineties at the exam. Which will be very difficult due to the time constraints anyway.

P.S.
Analysis. The difference between $\mathbf{R}$ and $\mathbf{Q}$ is where mathematics feels more like a creation than an invention. Did mathematics exist before humans populated the earth? Did we discover math or did we create it? This could lead to interesting thought or discussion. Riemann created a function which is continuous but nowhere differentiable. $$f(x)=\begin{cases}\frac{1}{q} \text{ if rational and }x=\frac{p}{q},(p,q)=1\\0 \text{ if irrational}\end{cases}$$

New Youtube Video

I made a tiny Mathematica program which demonstrates all the 23+1 rotations of the Cube. If you haven't done Group Theory yet: the Cube has four diagonals, which can be permuted in 4! = 24 = 23+1 ways. The 23+1 rotations create just these permutations. - I joined Facebook ( at last ), found an M208 study group but with disappointing little discussion. Here is the video:

M208 Group Theory video

To give you an idea what the OU videos ( delivered on DVD of course ) are like, I made some pictures.

This video is about the Counting Theorem. Applying this theorem enables you to answer questions like: "How many different dodecahedrons are there up to rotation if a face can have any of five colours?". - The OU videos are excellent. The counting theorem is one of the more difficult theorems in group theory. I don't think there is a better way of explaining it than they did in this video. - You can't compare an OU video with a lecture in class. The lecturer simply doesn't have the resources that are used in preparing a video like this. - Just think of the hundreds of abstract algebra lecturers over the world repeating the same talk they did last year. And of the same lesser quality than if they would work together and prepare videos like this one. I am sure we are heading to that direction. Just look at all the educational content that is already available for free on internet. Teachers don't have to be afraid of losing their jobs. They can create new materials, help students in small groups and do research.

Result M208 - TMA04

Result M208 - TMA04, 76%. I am not sure if I would have deserved more than 76%. Real Analysis AA was extremely boring. AB is less boring, btw. - I suppose that on universities the stuff in the various MST121, MS221 and M208 books is grouped and lectured by topic. My guess is four 30 point courses called Calculus I, Linear Algebra, Group Theory and Discrete Mathematics is the equivalent at a university. If you don''t do well in one subject all your results are dragged down considerably at the OU. At a university your Calculus grade would be less for example. - Next Action: Group Theory B. A topic I can relate with.

M208 GTB

Bookset 5 of M208 is about Group Theory again. The main topics are conjugacy, homomorphisms and group actions with counting. I started studying the material as I want to start working on the TMA if at all possible this week. I did make time for reading about some other, although closely related, topics though: matrix groups. There are some beautiful maps $f: \mathbb{C} \rightarrow GL_2(\mathbb{R})$ mapping complex numbers to equivalent real matrices. - I am close to getting the click / cognition about the essence of Lie Groups. More about that another time.

Cayley's Table

Although Evariste Galois laid open the route to Group Theory his paper was basically about the theory of algebraic equations. The first paper on Group Theory ( with a reference to Galois, of course ) was written by Arthur Cayley and published in the Philosophical Magazine, vol. VII. (1854), pp.40-47 and included in the Collected Papers of Cayley Vol. II. The article is called "On the theory of groups, as depending on the symbolic equation $\theta^n=1.$ If you have studied the M208 GTA books on Group Theory the paper is very accessible. My suggestion is that you read it. If you do you will certainly understand why group tables ( Cayley Tables ) are named after Cayley.

( Although we can learn all the required mathematics for B31 straight from the OU books I still want to know how the guys who created all that miraculous stuff originally wrote it down. Besides that there will come a day that we have to read original papers anyway. )

LU Factorization

$\left(
\begin{array}{ccc}
1 & 2 & 3 \\
2 & 6 & 10 \\
3 & 10 & 12
\end{array}
\right) =
\left(
\begin{array}{ccc}
1 & 0 & 0 \\
2 & 1 & 0 \\
3 & 2 & 1
\end{array}
\right) \cdot
\left(
\begin{array}{ccc}
1 & 2 & 3 \\
0 & 2 & 4 \\
0 & 0 & -5
\end{array}
\right)
$
or
$A=L.U$ where $A$ is a non-singular matrix and $L,U$ are respectively lower- and upper-triangular matrices.

Every non-singular matrix can be factorized in the product of a lower- and upper-triangular matrix. ( The factorization itself is trivial. )

M208 - TMA04

Just finished M208-TMA04. Real Analysis. More difficult than I thought but I managed to complete the TMA.

M208 -TMA03 Result is in

M208 - TMA03 ( Linear Algebra ) result is in.. Result: 90%. Still four TMA's to go. Two on Real Analysis, one on Group Theory and a revision TMA just before the final exam. TMA results sofar were all in the distinction zone, and increasing: 85, 88, 90. Comparable to MS221 last year. That would mean I am heading for a grade 2 pass.

How am I doing (3) - May

M208
- TMA01 85% ( part 1  77% , part 2 89% )
- TMA02 88%,
- TMA03 Awaiting result. Delivered after cut-off ;-) Expecting 78 (72 - 84)
- Studying TMA04. Somewhat behind schedule )

MT365
- CMA41 88%
- TMA01 79%
Problems with MT365. Will/must concentrate on CMA42 with cut-off day after tomorrow first. Then evaluate situation. I simply haven't studied enough on MT365. And I know why. More when evaluation has been done. - My main-focus is M208 because it is mainline mathematics, it's required for B31 Pure Mathematics and represents 60 points. Focus will remain on M208.

M208 - TMA03

Done. At last. :-( - I really like the M208 topics though. TMA03 was definitely a case of serious bad planning. I hope the tutor accepts and will mark the TMA. The paradox is that I think I might get somewhere between 75-90 points for this TMA. Have to see and wait. ( Next urgency is MST365 CMA42. )

M208 - TMA03

I have asked for a last minute extension of the cut-off date. While reviewing the draft TMA I found several issues that need to be addressed further. - And in the meantime the cut-off for MT365-CMA42 is dooming up on the horizon.